Series representation o f random mappings and their extension

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Series representation o f random mappings and their extension

Dan u llui iu Thang*. Tran Manli C u o n g

i - a c u ỉív o f M a lììcnia ĩics, H a n o i U n iv e rsity of Scie nce, i'NU. 3 3 4 N^^uycfiTrai S ir , ỉ ỉa n o i, Vietnam

Rcccived 28 I'cbruary 2009 VNU Journal o f Science, M alhem alics - Physics 25 (2009) 23 7 -2 4 8

A ỉis tr a c t. Ill tiiis paper, we introduce a method o f extendin'-; the d om ain o f a random m apping a d n u lliim 1ỈÌC series expansion. This method is based on ihc convcrgcnce o f certain random series. S om e conditions u nder which a random m ap p in g can be extended to apply lo all X - vaỉucd random variables will bo presented.

A M S S u h jc c t c la s s ijia n io n 2000: Primary G 0 / / ( i 5 ; Secondary: G 0 / Ỉ 1 1 , G 0 G 5 7 , 6 0 A " 3 7 , 37A55.

K c\'\vords ư nd p h ra se s : random operator, bounded random operator, d om ain o f extension, action on random inputs.

1. Stries representation of random mappings

I x t V' be separable mctric spaces. By a random m apping from .Y into Y w e mean a rule

*I> tlut assigns to each elem ent X G A" a unique Y - valued random variable X. Equivalently, it is a niapfing *i> : i i X A' sucii that for each fixed X G A \ the map <!>(., x ) : ÍÌ Y is measurable.

In tl\is point o f view, two niappinus 1 : Q X X V, *l>-> : i i X X —► V" define the same randuii mapping if for cach X G -V

= 2( j - , u ; ) a , s .

N olirg that the exceptional set can depend on X . In this ease, \vc say tiiat the random m apping 4>2 is

a m o l i f i c a t i o i i o f l l i e r a n d o m m a p p i n g J.

DcTicition 1.1 A random m a pping from .V into r is said to adm it the series expansion if there cxisti a scqiieiicc ( / , ,) 0Í determ inistic measurable m appinus from -V into Y (rep. from X into R )

a n d I s c q i i c i i c e ( a , i ) o f r c a l - v a l u c d r a n d o m v a r i a b l e s ( r e p . y - v a l i i c d r . v . ' s ) s u c h t h a t

CO

w h e n ihc series converges in Lq .

I n t h e e a s e t h e s e q u e n c e { ( i n) i ire i n d e p e n d e n t \ v c s a v t h a t a d m i t s a n i n d e p e n d e n t s e r i e s e x p a i s i o n .

' Corcsponding authors. Tcl. >844.38581135:

ỉí-nail: hungthang-dang'T/giiiail.com

237

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P ro p o sitio n 1.2 ỉ.ct hc (ĩ random o pcraĩor from X into V lỉỉìd suppose íỉìiií X is a ĩìa n a ch s p a c t' with ihe S h a u d er b a sis c -- (í‘ri)r^, 1 c o ìýỉt^ a ie basis í * a d m iis ihe s e r ie s expansion.

R e c a l l that, a r a n d o m mappiii^^ <1> is c a l l c d a r a m l o m o p e r a i o r i f ii ÌS l ỉ ì ì c a r a n d s t o c h a s i i c a l l y continuous, i.e.

Í * ( Ả ị X ị ỉ Ao.z'2) " r A2‘1^^2r V . r i . . i ' o 6 -Y, A j . A o G M,

a n d

.r - N o te that the e x c ep tio n a l s e t m a y d ep en d on A i , A2, -/ ]. r j .

Proof. F'or each X G A", w e have

238 D .H Thuỉìị^, T h í C iion^ / \ 'NU J o u rn a l o f Scỉcncc, híuỊÌìcmưỊics - ĩ^ỉivsics 25 (2009) 237-248

oc

X = ^ ( : r ,

71^1 Since is linear and stochasticallv continuous, \vc get

oc n-1 w here the series converges in L]y .

Put Qn — ( ^ n ) is a sequence o f y - v a lu e d and ( / „ ) o f detcm iinistic m easurable m appings from A" into Y . \Vc have

oc

X = ^

n ^ \

n

A random m apping from .V iiUo is calk'd a sMiiinclric G aussian random m apping if for each k £ N and for each finite scqiiencc o f -V X y * the - valued random variable

is sym m etric and Gaussian.

T h e o r e m 1.3 Let he a svĩnnieỉric sto ch a stica lly C O Ì Ì Ĩ Ì Ì Ì I Ì O Ỉ Ì S iỉaiissỉU ìì iiỉĩìdom m appifi^. Thcìỉ 'I*

adm its an in d ep end en t series e x p a m io n

'DC'

n - - l

where ( a „ ) is a seqiieuce o f tv a l-y a liic J Gciussiaỉì i j . d ruìỉdoììì ya riuhỉes a n d fjị : -V —^ K is co n iifw o u s (so is m easurable).

Proof. Let [] denote the closcd subspacc o f A2( n ) spanned by random variables y*). X G A', y* € Y * } . T hen [] is a separable Hilbert spacc and every elem ent o f [*1>] is a s jn im c tric Gaussian random variable, Let ( a „ ) is an orllionormal basis o f Since the sequence {(\n) is orthogonal,

s y m m e t r i c a n d G a u s s i a n , it i s a s e q u c n c c o f r c a l - v a l u c d G aussian i . i . d r a i ì d o i ì ì \ a r i a b l e s . N o w f o r e a c h

n , w e define a m apping f n : X —^ r by

f n X = Ị C\n[uj yi ^rụ)(ÌP[^^)- JQ

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D Ị Ị 7'hun^, T hí. Cỉionịị / V N lỉ J o u rn a l o f Science, h i a t h a n a ti c s - ỈHìysics 25 (2009) 2 3 7 -2 4 8 239

l ỉ e r c t h e ỉ ĩ o c ỉ i n e r ( ! ) e x i s t s b c c a u s c b y C a u c h y i n e q u a l i t >

\n„{uy\> x{u;)\\dP {uj) < ( 2 )

I ' i x T e A'. I'or cacli Ị/* e r * , 7/*) € [‘I>1 is e x p a n d e d i n t h e b a s i s ( o „ ) i n t h e f o r m

.L

f f an'I>J-(/P(^0- 'y'i

n = l ^

oc

^ ,y )

n-=ĩ

fV

w h e r e t h e s e r i e s c o n v c r i i c s i n /v2( í ỉ ) s o it i s c o n v e r g e n t i n p r o b a b i l i t y . S i n c e t h e s e q u e n c e ( a n / n ^ ) is a s e q u e n c e o f s v n i m c l r i c i n d e p e n d e n t y - v a l u e d r . v . ’s , b v t h e I t o - N i s i o t h e o r e m , w e c o n c l u d e t h a t

oo

' f x = a . s .

n=\

Finally, fixing /Ỉ, w e sliow that fn is continuous. Let (x/,)

c

.V such that l i u i ^ Xk = X. From (2) we have

11/.^'^- - f n x f < E\\<l>x, - ^I>:r||^

B y t h e a s s u m p l i o i i p - --- <I*.r a n d t h e f a c t t h a t i n (<!*] a l l lliC c o n v e r g e n c e i n L p ( i i ) , { p > 0 ) a r c c í Ị u i \ a l c n t . w e h a v e l i i n / : ' | ị ‘l>.ỉ7, - ’l>.r||^ 0 . ' I ' h e r e i b r c , l i i i u - / n X A - -■ o

Next, wc sliall be interested in possible extensions o f T heorem 1.3 to the case o f symmetric

s t a b l e r a n d o m m a p p i n g s .

I el <h h e '.i rnnriotii Iiirippin[3 fr o m X in to V is Kíúd to h e a s y m m e t r i c p s t a b l e r a n d o m m a p p i n g (SpS random mapping in short) if the real proccss vy^^)} defined on A" X Y * is symmetric p -

stable, ỉn tliis case, for each X € -V, '1>X is a y^-valucd SpS random variable.

L e t [<!'] d e n o t e t l i e c l o s c d s u b s p a c c o f / . o ( i i ) s p a n n e d b y r a n d o m v a r i a b l e s { ( ^ x , y * ) , x G

A', y* E V**}. Ii'(^ e [<1*1 then (Ệ is SpS so ihc ch.f. o f is o f llic form 0XỊ){—c|/p^}, w here c = c(^) is a non-negative num ber depcndinii oti T he length o f ^ denoted by ll^ll* is defined by

II^IU =

It is k n o w n t h a t ( s e e [1]).

L e m m a

ij The c o n v s p o m ic fic e ^ * is an F -norm on [‘I»J a n d in fa c t is a norm in the ca se p > 1.

ii) [Ị is a liĩìcur slih sp a ce o f each L r { i i ) , 0 < r .

liij I h e F - sp a c e [*I>] can be isom eirically em b ed d ed itUo so m e Lp{S^ /í).

T h e o r e m 1,4 Let b e S p S s to c h a stic a lly continuous random m a p p in g a n d su p p o se that [<Ễ] is isom etric to Ip. Then adm its an in d ep en d en t series expansion

oo

n = \

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w h e r e ( a „ ) is a s c q i i c ì ì c c o f ì v a l - v a l u c d S p S i . i . d r a ì ì d o ì ĩ ì V í ĩ r ĩ í ỉ b ỉ c s (Ịỉìiỉ f , , : A' V ịs c í ư ỉ Ị i n u o ỉ i s ( s o li i s m e a s u r a b l e ) .

P r o o f , ỉ . e t / h e a n i s o i n c t r v íVoni [<1>Ị o n t o Ip a n d . / P u t

/ ( ( < k r , i y * ) ) - / J ( - r . / y ^ ) G I , .

A t f i r s t , w e s h a l l s h o w t h a t { ( \ u ) i s a s c q L i c n c e o f r c a l - v a ỉ i i c c l S p S i.i.cl r a n d o m v a r i a b l e s . I n d e e d , t l i c

joint ch.f. f i t \. 12... /„) o f the random va/iablc ( a Ị . í i ) ...is equal to

... ^ , ) = / ' r t ' X ] ) | y / V i ' x p | / ^ / a. / ( ( A . ) |

2*10 D ỉ ỉ Thdììị:^, T S Í Ciỉoỉỉi^ ' ./o u m u ỉ oJ'Sc'icfuv, M a ih cm a tic s - rỉìỴSỉcs 25 (2009) 237-24S

~ E e x p < i j

Ì Í i

^Ị ^{/ /Í} ^\

\ / c = l /

J I

^{V/, 1} ^J

a s d e s i r e d .

F o r e a c h { x , y * } G -V X V * , w e h a v e

(ijl ./(((()•

h e n c e

oc

n 1

where bri{x, y*) is the 7/-lh coordinate o f U ( x \ y " ) c Iị, and the series (3) c o in c tu c s in the norm ||.|Ị.

s o c o n v e r g e s i n p r o b a b i l i t y .

F i x 7/. W e s h o w t h a t t h e r e e x i s t s a m a p p i n g f „ ; X r s u c h t h a t f o r e a c h . r G x , y* G V "

F i x X € X . S i n c c t h e m a p p i n g y* (X, ;(/*) is l i n e a r s o t h e m a p p i n g ỉy" B { j \ y * ) i s l i n e a r w h i c h i m p l i e s t h e m a p p i n g bj: : y* * h n { x , y * ) f r o m i n t o R is l i n e a r . I n a d d i t i o n , t h e c h . f . o f is

r(V '*, y^)- continuous on V'*, w here r ( V*, 1") is the topolouy o f uniforni convergence on compact sets

o f y , a n d it is e q u a l t o

H A i r ) - e x p { - | | ( i > x , y ^ ) | i i ' } e x i ) { ~ p j ( x N / y * ) l i n

C o n s e q u e n t l y , bj: : r * —* R i s l i n e a r a n d r ( V * , V ) - c o n t i n u o u s o n y * . S i n c c l l i c d u a l s p a c c o f y * u n d e r t h e t o p o l o u y r ( V * , v^) i s >' \ v c c o n c l u d e t h a t t h e r e e x i s t s a u n i q u e e l e m e n t d e n o t e d b y J n X s u c h t h a t

= (/nX,iy*) "-■ (/-!■'% y*)-

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Novn . the cquaiilN (-■]] b c c o n i c s

/ ) ỊỊ 'ỈIuỉỉVì;. T.M. Cỉi()}\\ỉ, I'S l- J o u rn a l of Slìcììcc. S í a t h a n a ii c s - rỉìv.sics 25 (2()09) 237-248 2-11

/y*) - Ỵ ^ a J ) n Ự , Ị Ị " )

„ r . _ ỉ

:x:

/y’ ).

n--\

ỉ he rest oị' p ro o f is carried out similarly as in the p ro o f o f 1 hcorcm 1.3

Ị inall>, llxiriiz it. \vc show tlial fn is continuous. Let be a scquencc o f A" such that liin./ Ẳ . B \ the assum ption Ị)-liin \vc have

Ả*

oo

- <1>X zrz - f j x ) .

S ince Ị) < '2 by Corrolarv 7.3.G iti [2 , we get

\ \ fn. r, - f n . > r < E - / r ' i l " < C { m > . r , -

7 i

w h e r e / - < /; a n d t h e c o n s t a i i l c > 0 d e p e n d s o n l y o n r , / > . I ' r o i i i 2. o f L c i i i i i i a \ v c o b t a i n 1ì i i i^. {F| | Xẳ: —

.r|p'} ^ Í). Hence. f , , r as desired. □

2. I he e x te n s io n o f r a n d o m m a p p in g s u d m ittin g series e \|):u is io n

l et *1' be a randoni inappinu IÌ0I1Ì .V into r adiniltiiìíí llie s er i e s e x p a n s i o n

(^1) where (/,() is a scqucncc o f dclcrniinistic measurable inappinus froiii A" into y (rep. from A" into R ) and ( o „ ) is a secỊuence o f real - valued random variable (top. V' - valued r.v.). riic series converges m >.

Denote In P(<1>) the set o f all X - valued r.v. it such lliat tlie series oc

(^nĩn^ỉ (5)

n = l

coỉìvcrgcs in probabilit\. Here /,ií/(^ ') ~ /;, (u(u.’)) is a raiuloni variable because / n is measurable.

Clearly, X

c

P()

c

^{L Ì Ỉ}.

D efin itio n 2.1 is callcd the dom ain o f extension o f <1>. If ỈI G P ( '1 0 then the sum (5) is denoted by ‘I‘í/ and it is understood as tlic aclioii o f on the random variable u.

T h e o r e m 2.2 ( f u is a c o im ta b ly - valu ed r \\

oc

u = ^ Ie.x,

i = l

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w h ere [ E ^ , i 1,2, . . . ) is a c o u n ta b le p a r iitio n o f i i a u d A\ G , v . th en II e P ( ' l ' ) (ỉĩĩd

<l>a = ^ .--I

Proof. Pul Z n ^ X] " = 1 «i/iW and z = We have to show tliat

\ \ m P { \ \ Z n - Z \ \ > / ) = ( ) . n

S i n c e UJ ^ E k ^ ~ Z n { o j ) ~ c i i f i X k s o

1=1

P(||Z„ - Z|| > 0 = Ễ - ^li >

k=:\

jV / n

A-=l

V

^{t = l}

242 D .//. Thaug, T M . C u o n g / \'N U J o u r n a l o f S c ia w e . híưiỉìcm aỉics - Physics 25 {2009) 2 3 7 -2 4 8

< > t

OC:

For each k — 1, 2 , A" w e have

71

l i n i P d l - <I'Xi.|| > t ) =

1 = 1

0.

Let n —> oo and then N —►CO, w e get liiii,, P {\\z ,^ - Zịị > t) = 0. □ For each random m a pping adm itting the representation (4), let J ' ( n ) denote the (7-algebra generated by the family { a , J . A random variable u e Lq is said to be independent of if and T { a ) are independent.

T h e o r e m 2.3 S u p p o se th a t II is iiidependent o f , then u €

Proof. Let t > 0. By the independence o f u and the sequence ( n „ ) \vc have p

/ n \ r / 71 \

> t

=

/ /

^{> /}

\

^{i —rn}

)

^{J x}

V

^{Ĩ — ÌÌÍ} ^/

w here fi is the distribution o f ĨI. Because for each :r G A' liin p

ĩìiji~*oo

/ n

By the dom inated convcrgcnce theorem, w e infer that / ri liin p

in ,n —*oc a . / . u i l > t

\ i=rn

= 0. /

Therefore, the series

oo

t = l

□

converges in L ị ' i.e. ii ĩ>(í>).

T h e o r e m 2.4 L ei

^

j e a random m a p p in g fr o m

X

into

y

the series expansion o f the fo r m (4). Su p p o se th ' <

c

f o r a ll k, w here p > I a n d q is the co n ju g a te n u m b e r o f p (i.e.

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ỉ ) ỉ ỉ Thung, T.M. Cu(yn\i / VNU J o u rn a l o f Science. M a ỉh e m a iic s - Phỵsics 25 (2009) 23 7 -2 4 8 213

^Ị) t i /V/ - lÁ F o r ư € L(Ỷ ío h c lo n ^ to V{i *), a su fficie n t co n d itio n is

< oo.

Proiif. Pul

A p p l y i n u t h e ỉ l ồ l d c r i n c q i i a l i l v , w c g e t

E

= m i h v r }

n ri

^ . ru-A-ii < ^ £'lafc|||/A,.!i|

V —rn k — in

n

k~ni n

< c ^ ri,{q) —* 0 as m , ii —» oc.

k ~ r n

(

6

)

PC

□

l l c i i c c , the s er i e s

Y"

c o n v e r g e s in L \ s o c o n v e r uc s in

L ị .

k-~\

C o r r o l a r \ ' 2 . 5 S u p p o s e t h a t i s a s y m m e t r i c s i o c h a s i i c a l l v c o i i t i m i o u s G a u s s i i i f i r c w d o m m a p p i n g

and if

Y ^ { E \ \ { h u W Y / ' >

k fo r so m e q > 1 tiicn u G

3. W hen a random m a p p in g can be extended to the entire space

l.ci 'I' be a laiKliHit u p c i a t u i i'u)ii» A' iiilu V a n d bii[>pu5c llial A is a s c p a i a b l c I3aiiach s p a c c w i i h t h e S h a u d c r b a s i s i ' - Ị i i n d t h e c o n j u g a t e b a s i s ( * (^n)ÍTỈ^[* P r o p o s i t i o n 1 . 2 ,

a d m i t s t h e s c r i e s e x p a n s i o n .

oc

71^ i I hcorcm 3.1

II / / is a b o u n d e d r a n d o m o p e r a t o r t h e n a n d *1>U d o e s f w t d e p e n d o n t h e b a s i s {f'n)-

ii) C o n v e r s e l y . / / P ( * l > ) — t h e n m u s t h e a h o u n d e d r a n d o m o p e r a t o r .

R c c i i l l ílìiiỉ a r c m d o m o p e r a t o r 4> is sa id t o h e h o u n d e d i f i h e r c e x i s t s a r e a l - v a l u e d

random v a r i a b l e k { u : ) s u c h t h a t f o r e a c h X ^{G -V}

| ‘I>:r(u.’)l| ^ / l ' ( c j ) | | x | | a.s,

Noiiỉì}^ that the exceptional set may depend on X.

P r o o f : i) S i n c e <1> is b o u n d e d , b y T h e o r e m 3 . 1 [3] t h e - e e x i s t s a m a p p i n g

T : Q ^ L { X , Y )

s u c h t h a t f o r e a c h X e -Y

4 > x ( u ; ) — T {lu)x a . s .

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2‘t‘l D.ỊỊ. 77ỉí//íi;, T.SÍ Cmmị:^ / \ 'NU JouniLil ofScivnce. S í ư Ị Ĩ ì c m a í i c s - ỉ ^ h y s i c s 25 ( 2009} 237-24S

As a conscquciicc, thcic is a set D wiili F ( D ) ~ 1 such llial ỉbr cach G and for all ÌI \vc have

Hence, for each e D

D C DC

' ^ ( u( uj ) , c *, y i >e n{ uj ) = Ỵ 2 [ l l { a . ') . c * j r ụ ) r n

n~ \ n^-1

/ ^ \

\ n = \ /

Therefore, the series (ỉ/, converges a.s. so coĩìverues in probability. Consequently;

u e 'D(í>) and ’u(u;) = T{ í j ) { u{ uj ) ) does not depend on tlie basis c ~ (i"n)- ii) Put

71 1=1

Then is a linear c o n t i n u o u s m a p p i n g from Lg into Ay . B v the assLiniplion l i i u, i for a l l a G Lq . I l e i i c c , b y t h e B a n a c l i - S t c i n h a u s t h e o r e m ‘1> is a u a i i i a l i n e a r c o t u i n u o i i s m a p p i n g f r o m

Lq into L ị . In addition, we have

“ Z] w here { E i , i — 1, is a partition o f ÍÌ and :rj e X . By Theorem 5.3 |3 wc 1-1

concludc that i s bounded. □

T h e o r e m 3.2 L ei Ỷ he a random o p era to r a d m iíù ìĩ^ ihc series cxpaììsioìì of' ific form (4), where (o,() is a sequence o f rea l-va lu ed ram iom variables am i

(/„)

is a scqucucc o f COỈÌÙÌÌUOUS linear ĩìĩappings from X into V'. Thcii

i) 7/‘ is b o u iu led then P() ^ Lq .

ii) Conversely, i f V{ A^ ) ” L;} then m u sĩ be hoim ded.

Proof\\) Sincc i> is bounded, by Theorem 3.1 [3] there exists a niappinii

T L { X , Y )

such that for each X ^{G -V}

= T{;ijj)x a.S.

For t h i s reason, t h e r e i s a s e t D w i t h F { D ) — 1 SUCÍ1 t h a t f o r c a c l i uJ E D a n d f o r al l k vvc h a v e

‘I>a-(í^) = 'ỵ ^ a n { i^ )fn C tc = T{uj)eh.

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D i Ị. i.M. iAion*^ / \'N U J o u rn a l o f Science, M a th e m a tic s - Physics 25 (2Q09) 237-248 245

As ía c(Miseqiiencc% for each u: e D

= J ^ a n { u j ) f n ( Ỵ 2 < > (^-k)

it n k

= (hii'-j) < u{u'), c ị > fnCk

il k

k- n

^ < u{uj),el. > T[iú)ck k

/ ^ \

= r { u i ) ^ < u{uj), c l > Ck

V k /

ii) F^ut

ri 1 = 1

'Ị lien is a linear co n tin u o u s m apping from Lq into Lq . Bv the assum ption linia — U for all ỉi G Hence, bv tlic Banach - Sleinhaus theorem is again a linear continuous mappiníĩ from

i i i t o Lịị . h i a d d i i i o n , f o r ii - X l i L i w h e r e { Fi ^ i — 1, is a p a r t i t i o n o f í ỉ a n d Xj E -V, we h a \ e

oc-

k 1 (X-' IÍ

1=1 Tl

I- 1

ÌÌ 'V^

1/.;,

1=1 k=\

ỉỉy riicorciu 5.3 [3] u c c o n c lu d e that is bounded. □

I'heoreni 3.3 Lcl X he a co m p a ct m etric sp a ce a n d <Ị> he a random m a p p in g fr o m A" inỉo Y adm iting the scries e x p a m io n o j th e fo rm (4), where ( o n ) is a seq u en ce o f fv a l-v a lu e d sym m etric itiJepem lent random variables ìỉỉìd ( / n ) is a seq u en ce o f co n iin u o u s ĩtuỉppings fr o m X into V.

i) //^I* has a co ììỉin u o u s m o dijica lion then e v e ty u G Lq b elo n g s to P(*P) i.e. = Lq . iij H ie convcr.se is ĨÌOÍ iru e i-C. there exists a ra rd o m m appỉììịị ft'o m X ink) >' a dm iiing ỊỈÌC s c n e s exp a n sio n o f (he fo r m (4), where (a ,i) is a seq u en ce o f rea l-va lu ed sym m eiric independent rafhlam variables a n d ( J n) seq u en ce o f ccriiinuous mappings from X into V such that P(4*) == Lq but <1» //Í/.V fU)i a co n lin u o iis ỉnoiiựìcaíion.

Proof. Let V' ” C ( A \ y ' ) be the set o f all continuous m appings from A" into y'. It is known that r is a separable Banach space under the suprem um norm

I / 1 K' = i^'ip II/(j-')

x€Ã'

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F o r c a c h p a i r (./■, ụ*) € -V X Y * t h e m a p p i n g X 2 ụ* ; V' R g i v e n b \

( . r c ạ y * ) ( / ) = ( / ( x ) , y * )

is d e a r l y an elem ent o f Let r — {(j; ® y*), (x,/y*) e -V X }' *}. ỈI is easy to check that 1' is a separating subset o f V*. Let 'p(x,íxj) be a continuous modification or<l*. D cllnc a niappinii T V by

T { u : ) ~ X ^

We show that T is m easurable i.c T is a V -valued random variable. iiKÌecci, ibr eacli (./’ 0 y*) G r t h e m a p p i n g w ' ( 7 " ( u . ’), X ộ<) 7y^) = ( 7 ’( c j) x , ?y*) = //") a.s. is m e a s u r a b l e . S i n c c I " is s e p a r a b l e a n d r is a s e p a r a t i n g s u b s e t o f \ t h e c l a i m s f o l l o w s f r o m l l i c t h e o r e m l . l i n ( H D -

N o t e that for c a c h

u

the m a p p i n g

X an{u)fjiX

is an e l e m e n t o f \ i ỉ c n c c

(infn

is a I ' - v a l u e d r.v. N ow for each ( x 0 y * ) € r we have

{ T { u j ) , x ® y * ) = { r { u ) x , i / ) = (x(a;),y*)

oo oo

= ^ ( a „ ( c j ) / „ a : , y ' ) = Y ^ { a n { u .') fn , x « y*) a.s.

n = l n = l

Since is a sequence o f y - v a lu e d symmetric independent r.v.’s in view o f Ito - Nisio theorem

oo

we conclude that the series <^n{^)f n converges a.s. to T ill the norm o f V'. 1'liis im plies that there n=]

exists a set D o f probability one such that for each UỈ £ D , x e -V, we have

T { u ) x = ^ a „ (L v )/n X .

Consequently, for II € Lq we have

7\ uj ) { u{ u; ) ) ^ ^ o „ ( c j ) / „ ( ' u ( a ; ) = ^ Q ,i(a;)/n »(a)) Vu; e I)

U=1 n=l

i.e. the series aniu>)fnu{iL>) converges a.s.

ii)The following example shows that the converse is not true.

Example. Let A' = [0; 1], Y = R. Consider the sequence (^n) o f real-valued independent r.v.’s given by

= - n ) = P {^n = ” ) = ^ - = <>) = 1 - Ậ - Then (^„) are real-valued sym m etric independent r.v.’s and

E(6.) = 0,E|Cn| = ^ , E | 6 . P = 1.

Let (a„) be sequence o f positive numbers defined by 1

2-ÍG D.ỉỉ. 7'hiinịỊ, T.\í. C u o n ^ / VNU Jou rn a l o fS c ie tw e , Síaỉhernatics - ỉ^hvsics 25 {^ìOOì 237-24S

( I n —

>/n log2 n

and put a „ = a„^„. Then (q„) are real-valued symmetric independent r.v.’s and E ( a „ ) = 0, E | q „ | = — , E | a „ | ^ = a ị .

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l.ct {f.^) be the s e q u en ce o f functions f n : [(); 1| —> K dcfitied b\' / „ ( / ) = coh27ĩ7/^.

Cleaarly, / „ are coDlinuous. C onsider the random function : [0; 1] —^ R given by

Ị) //, I'han'^.

TSÍ.

Ciionsị /

\

^'

su

J o u rn a l o f Science, M cuhem atics - P hysics

25 (2009) 237-248

²⁴⁷

oo

n = l

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\Vc ! have

r»C' oc oo

^ E | o „ / „ ( 0 | < ^ E | a „ | = ^ < oo

TI ~ I n = l n = Ỉ

siiicce jir < oc,

aị ~ Y1

^I*'⁵ ^<CO- I his implies the scries (7) converges a.s. Moreover, for

/i--1 n—\ n-~]

e ac h i real - valued random variable u we have

Thiss im plies the series

oc DC' CX5

y E \ a , J n { u ) \ < y E | a „ | = y ] ^ O c .

/ ^ / Z — y J-Ị

n = ) n = i n = l

oc

n-=]

c o n w e r g e s a . s. H e n c e P { ‘I») ^ / > o ( K ) -

Next, w c shall show that is an unbounded function. 1'o this end, w e use the following rcsuiili iroin ([5]) (Tlieorcni 7 and Hxersise 3 p. 231).

C onsider th e random scries

oo

a - ( f ) M = e [0;1

n = l

w h c r r e ( ^ „ ) a r c i n d e p e n d e n t a n d S N i i i m c l r i c r. v. ’s w i t h E | ^ „ p = 1, ( « „ ) a r c p o s i t i v e real n u m b e r s s u c h

tiiat y ^ n ^ 'n < CO and / „ ( / ) = c o s 2 n i i t . Put

/ \ i / 2

\ 2 ‘<n<2*ti /

•x

1 hctn if Si oc then is not a bounded function on [0; Ij a.s.

N o w w e com e back to our example. \Vc have

•'’. =

E

^«n > ( 2 ‘a ị , , ) " ■ ' =

\ 2 ‘<n<2'<> / 1)

cc

wliicch implies that Y2 '‘^1 “ riicrcforc, for almost sure UJ, i>(/)(u;) is not bounded a.s. so is not

Ỉ - 0

contiiiiuous on [0; 1 a.s.

A c k m o w lc d g m e n t. This work was supported in part by the Vietnam N ational F-oiinciation for Science and Technology Developm ent.

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248 D.ỉi. Thaiìs^, T.M. CAiouịị / VNU J o u rn a l o f Science. M a th e m a tic s - P h ysics 25 (2009} 2Ĩ7-24H

R cfcrc n ccs

111 lirclagnollc. J . Dacunha - Castcllc, Í). and Krivinc, J. L. (1966). Lois stable ct csp.ico Lf,, Ann. ỉnsĩ. // i\ìincarủ, ỈÌ2, 231 - 259.

|2 | \V,Ị jndc, infinitely divisible and stable measures on Banach spaces, Ixip/.ig 1983.

|3 ] D .II.'Ih an g and N .l'h iiih, R an d o m bounded operators and ihcir extension, K y u sh u J. Siiith. \'ol 5S(2()()4). 257-276.

|4 | N,N. Vakhania. VI, Tariclad/.e and s.A. Chohanian. i^robability Distrihiition on Banach spaccs. D.Rcidcl i’ublishing C'ompan). Dordrcchl 19X7

[5] J, P. Kahane. Some random series o f Junctions. CaiiibriJgc Univ. Press 1985.